5. Fluctuations and Second Order Freeness

Abstract

Given an N × N random matrix ensemble, we often want to know, in addition to its limiting eigenvalue distribution, how the eigenvalues fluctuate around the limit. This is important in random matrix theory because in many ensembles, the eigenvalues exhibit repulsion, and this feature is often important in applications (see, e.g. [112]). If we take a diagonal random matrix ensemble with independent entries, then the eigenvalues are just the diagonal entries of the matrix and by independence do not exhibit any repulsion. If we take a self-adjoint ensemble with independent entries, i.e. the Wigner ensemble, the eigenvalues are not independent and appear to spread evenly, i.e. there are few bald spots and there is much less clumping; see Fig. 5.1. For some simple ensembles, one can obtain exact formulas measuring this repulsion, i.e. the two-point correlation functions; unfortunately these exact expressions are usually rather complicated. However, just as in the case of the eigenvalue distributions themselves, the large N limit of these distributions is much simpler and can be analysed.